cw formula begin

article_de_rham_cyclic.bbl

@@ -44,7 +44,7 @@ N.~Dummigan.
 \newblock {\em Compositio Math.}, 119(2):111--132, 1999.
 
 \bibitem{Ellingsrud_Lonsted_Equivariant_Lefschetz}
-G.~Ellingsrud and K.~L\o~nsted.
+G.~Ellingsrud and K.~L{\o}nsted.
 \newblock An equivariant {L}efschetz formula for finite reductive groups.
 \newblock {\em Math. Ann.}, 251(3):253--261, 1980.
 

article_de_rham_cyclic.out

@@ -1,5 +1,5 @@
 \BOOKMARK [1][-]{section.1}{\376\377\0001\000.\000\040\000I\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n}{}% 1
-\BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000N\000o\000t\000a\000t\000i\000o\000n}{}% 2
+\BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000N\000o\000t\000a\000t\000i\000o\000n\000\040\000a\000n\000d\000\040\000p\000r\000e\000l\000i\000m\000i\000n\000a\000r\000i\000e\000s}{}% 2
 \BOOKMARK [1][-]{section.3}{\376\377\0003\000.\000\040\000C\000y\000c\000l\000i\000c\000\040\000c\000o\000v\000e\000r\000s}{}% 3
 \BOOKMARK [1][-]{section.4}{\376\377\0004\000.\000\040\000P\000r\000o\000o\000f\000\040\000o\000f\000\040\000M\000a\000i\000n\000\040\000T\000h\000e\000o\000r\000e\000m}{}% 4
 \BOOKMARK [1][-]{section.5}{\376\377\0005\000.\000\040\000E\000x\000a\000m\000p\000l\000e\000s}{}% 5

article_de_rham_cyclic.tex

@@ -167,7 +167,7 @@ In the second step, we use similar methods to show the result for a group of the
 $\ZZ/p^n \rtimes_{\chi} \ZZ/c$.  Finally, we use Conlon induction theorem to deduce Main Theorem
 for an arbitrary group with a cyclic $p$-Sylow subgroup.
 %
-\section{Notation}
+\section{Notation and preliminaries}
 %
 Assume that $\pi : X \to Y$ is a $G$-cover of smooth projective curves over an field $k$
 of characteristic $p$.
@@ -190,7 +190,6 @@ Throughout the paper we will use the following notation for any $P \in X(\ol k)$
 \end{itemize}
 %
 By Hasse--Arf theorem (cf. ???), if the $p$-Sylow subgroup of $G$ is abelian, the numbers $u_{X/Y, P}^{(t)}$ are integers.
-
 For any $Q \in Y(\ol k)$ we denote also by abuse of notation $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
 $u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.
 Let
@@ -199,9 +198,45 @@ Let
 	B_{X/Y} := \{ Q \in Y(\ol k) : e_{X/Y, Q} > 1 \}
 \]
 %
-be the branch locus of $\pi$. Recall that $\ZZ/p^n$ has $p^n$ indecomposable representations over a field of characteristic~$p$.
+be the branch locus of $\pi$. In the article we often use the Iverson bracket notation:
-We denote them by $J_1, \ldots, J_{p^n}$. Observe that $J_i$ is given by the Jordan block of size $i$ and eigenvalue $1$.
 %
+\[
+\llbracket P \rrbracket = 
+\begin{cases}
+	1, & \textrm{ if $P$ is true,}\\
+	0, & \textrm{ if $P$ is false.}
+\end{cases}
+\]
+%
+We review now some facts from representation theory of finite groups.
+Recall that $\ZZ/p^n$ has $p^n$ indecomposable representations over a field of characteristic~$p$.
+We denote them by $J_1, \ldots, J_{p^n}$. Observe that $J_i$ is given by the Jordan block of size $i$ and eigenvalue $1$. Assume now that $G$ is a finite group with a normal cyclic $p$-Sylow subgroup $H = \langle \sigma \rangle \cong \ZZ/p^n$. Let $C := G/H$.
+Recall that if $U$ is an indecomposable $k[G]$-module
+then $U^{\sigma} := \ker(\sigma - 1)$ (the socle of $U$) is an indecomposable
+$k[C]$-module. It turns out that the map
+%
+\begin{align*}
+	\Indec(k[G]) \to \Indec(k[C]) \times \{ 1, \ldots, p^n \}\\
+	U \mapsto \left(U^{\sigma}, \frac{\dim_k U}{\dim_k U^{\sigma}} \right)
+\end{align*}
+%
+is a bijection (cf. \cite[p. 35--37, 42 -- 43]{Alperin_local_rep}). We write
+$\mc V(M, i)$ for the $k[G]$-module corresponding to a pair $(M, i) \in \Indec(k[C]) \times \{ 1, \ldots, p^n \}$.
+
+Finally, we recall the classical Chevalley-Weil formula. Keep the above notation and assume that $p \nmid \# G$. For any $Q \in Y(k)$ let $\chi_Q : G_Q \to k^{\times}$ be the fundamental character of $G_Q$ acting on the tangent space of $Q$. Then:
+%
+\begin{equation}
+	H^0(X, \Omega_X) \cong \bigoplus_{W \in \Indec(k[G])} M^{\oplus a_M},
+\end{equation}
+%
+where:
+%
+\begin{align*}
+	a_M := - \dim_k M + \sum_{Q \in Y(k)} \sum_{i = 0}^{e_{X/Y, Q} - 1} \left\langle \frac{-i}{e_{X/Y, Q}} \right\rangle \cdot N_{P, i}(M),
+\end{align*}
+%
+and $N_{P, i}(M) := ???$.
+
 \section{Cyclic covers}
 %
 \begin{Theorem} \label{thm:cyclic_de_rham}
@@ -237,20 +272,9 @@ but it works for an arbitrary module).
 Moreover, for $i > 0$:
 %
 \begin{equation} \label{eqn:dim_of_Ti_Jl}
-	\dim_k T^i J_l = \llbracket i \le l \rrbracket,
+	\dim_k T^i J_l = \llbracket i \le l \rrbracket.
 \end{equation}
 %
-where we use the Iverson bracket notation:
-%
-\[
-\llbracket P \rrbracket = 
-\begin{cases}
-	1, & \textrm{ if $P$ is true,}\\
-	0, & \textrm{ if $P$ is false.}
-\end{cases}
-\]
-%
-
 In the inductive step we use also the group $H' := \ZZ/p^{n-1}$. In this case
 we denote the indecomposable $k[H']$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$
 and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.
@@ -308,6 +332,7 @@ The equality $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ does not hold for non-cyclic
 		\wedge(x_{1}, \ldots, x_s) \otimes \mathbb{F}_p[x_1, \ldots,x_s] & \text{ if } p>2
 	\end{cases}
 \]
+Therefore, for $s>1$ the degree one and two parts of the cohomological ring, which correspond to the first and second cohomology groups, have different dimensions. 
 \end{Remark}
 }
 %
@@ -420,15 +445,15 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 	Recall also that by \cite[???]{Serre1979} there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that for every $t \ge 0$:
 	%
 	\begin{align*}
-		u_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} + \ldots + i_{X/Y, P}^{(t-1)}\\
+		u_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} + \cdots + i_{X/Y, P}^{(t-1)}\\
-		l_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \ldots + i_{X/Y, P}^{(t-1)} \cdot p^{t-1}.
+		l_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \cdots + i_{X/Y, P}^{(t-1)} \cdot p^{t-1}.
 	\end{align*}
 	%
 	Observe that:
 	%
 	\begin{align*}
 		i_{X/X', P}^{(0)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p,\\
-		i_{X/X', P}^{(t)} &= p \cdot (i_{X/Y, P}^{(t + 1)} + \ldots + i_{X/Y, P}^{(n-1)}) \quad \textrm{ for } t > 0.
+		i_{X/X', P}^{(t)} &= p \cdot (i_{X/Y, P}^{(t + 1)} + \cdots + i_{X/Y, P}^{(n-1)}) \quad \textrm{ for } t > 0.
 	\end{align*}
 	%
 	This implies that
@@ -441,10 +466,10 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 	%
 	\begin{align*}
 		p \cdot (u_{X/Y, Q} - 1) &=
-		p \cdot (i^{(0)}_{X/Y, Q} + \ldots + i^{(m_Q - 1)}_{X/Y, Q} - 1)\\
+		p \cdot (i^{(0)}_{X/Y, Q} + \cdots + i^{(m_Q - 1)}_{X/Y, Q} - 1)\\
 		&= (p-1) \cdot (i^{(0)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y, Q} + p \cdot i^{(1)}) \\
-		&+ p \cdot (i^{(2)}_{X/Y, Q} + i^{(3)}_{X/Y, Q} + \ldots) - 1 \\
+		&+ p \cdot (i^{(2)}_{X/Y, Q} + i^{(3)}_{X/Y, Q} + \cdots) - 1 \\
-		&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y', Q'} + i^{(1)}_{X/Y', Q'} + \ldots - 1)\\
+		&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y', Q'} + i^{(1)}_{X/Y', Q'} + \cdots - 1)\\
 		&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (u_{X/Y', Q'} - 1).
 	\end{align*}
 	%
@@ -584,32 +609,10 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 	is uniquely determined by the $k[C]$-structure of $T^1 M, \ldots, T^{p^n} M$.
 \end{Lemma}
 \begin{proof}
-	This is basically \cite[proof of Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. We sketch the proof for reader's convenience. 
+	This is basically \cite[proof of Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. We sketch the proof for reader's convenience. Let $\psi : C \to k^{\times}$ be a primitive character. Write
-	Assume that $G$ is a finite group with a normal cyclic $p$-Sylow subgroup $H = \langle \sigma \rangle \cong \ZZ/p^n$. Let $C := G/H$.
-	Recall that if $U$ is an indecomposable $k[G]$-module
-	then $U^{\sigma} := \ker(\sigma - 1)$ (the socle of $U$) is an indecomposable
-	$k[C]$-module. It turns out that the map
-	%
-	\begin{align*}
-		\Indec(k[G]) \to \Indec(k[C]) \times \{ 1, \ldots, p^n \}\\
-		U \mapsto (U^{\sigma}, \frac{\dim_k U}{\dim_k U^{\sigma}})
-	\end{align*}
-	%
-	is a bijection (cf. \cite[p. 35--37, 42 -- 43]{Alperin_local_rep}). We write
-	$\mc V(M, i)$ for the $k[G]$-module corresponding to a pair $(M, i) \in \Indec(k[C]) \times \{ 1, \ldots, p^n \}$.
-	Thus it comes from a character $\chi_U \in \wh{C} := \Hom(C, \CC)$.
-	It turns out that the map
 	%
 	\[
-		U \mapsto (\dim_k U, \chi_U)
+		M \cong \bigoplus_{a, b} \mc V(\psi^a, b)^{\oplus n(a, b)}.
-	\]
-	%
-	is a bijection between the set of indecomposable $k[G]$-modules and the set $\{ 1, \ldots, p^n - 1 \} \times \wh{C}$. Fix a character $\chi$ that generates $\wh{C}$.
-	Write $U_{a, b}$ for the indecomposable $k[G]$-module with socle $\chi^a$
-	and dimension $b$. Write
-	%
-	\[
-		M \cong \bigoplus_{a, b} M_{a, b}^{\oplus n(a, b)}.
 	\]
 \end{proof}
 %
@@ -692,18 +695,18 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 \begin{proof}[Proof of Main Theorem]
 	As explained at the beginning of this section, it suffices to show this in the case when $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$ and $k = \ol k$ by Lemma~\ref{lem:reductions}.
 	We prove this by induction on~$n$. If $n = 0$, then it follows by Chevalley--Weil theorem.
-	Consider now two cases. Firstly, we assume that $X \to Y$ is \'{e}tale. Then the results of \cite{Garnek_equivariant} imply that the Hodge--de Rham exact sequence splits and
+	Consider now two cases. Firstly, we assume that $X \to Y$ is \'{e}tale. Then by Lemma~\ref{lem:G_invariants_\'{e}tale} and \cite[Corollary~2.4]{Garnek_equivariant} we have $\dim_k H^1_{dR}(X)^H = 2g_Y = \dim_k H^0(X, \Omega_X)^H + \dim_k H^1(X, \mc O_X)^H$. Therefore the Hodge--de Rham exact sequence splits by \cite[Lemma~5.3]{Garnek_equivariant} and
 	%
-	\begin{equation}
+	\begin{align*}
-		H^1_{dR}(X) \cong H^0(X, \Omega_X) \oplus H^1(X, \mc O_X).
+		H^1_{dR}(X) &\cong H^0(X, \Omega_X) \oplus H^1(X, \mc O_X)\\
-	\end{equation}
+		&\cong H^0(X, \Omega_X) \oplus H^1(X, \mc O_X)^{\vee}
+	\end{align*}
 	%
-	By Lemma~\ref{lem:G_invariants_\'{e}tale} and \cite[Corollary~2.4]{Garnek_equivariant} we have $\dim_k H^1_{dR}(X)^H = 2g_Y = \dim_k H^0(X, \Omega_X)^H + \dim_k H^1(X, \mc O_X)^H$. Therefore the Hodge--de Rham exact sequence splits by \cite[Lemma~5.3]{Garnek_equivariant}.
+	(the last isomorphism follows from Serre's duality, cf. ???).
-	Recall that by~\cite[Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}
+	Now it suffices to note that by~\cite[Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}
-	the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the higher ramification data. This ends the proof in this case, 
+	the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the higher ramification data. This ends the proof in this case.\\
-	as $H^1(X, \mc O_X) \cong H^0(X, \Omega_X)^{\vee}$ by Serre's duality (cf. ???).\\
 	
-	Assume now that $X \to Y$ is not \'{e}tale. Analogously as in the previous case, Lemma~\ref{lem:TiM_isomorphism_hypoelementary} and proof of Theorem~\ref{thm:cyclic_de_rham}
+	Assume now that $X \to Y$ is not \'{e}tale. Lemma~\ref{lem:TiM_isomorphism_hypoelementary} and proof of Theorem~\ref{thm:cyclic_de_rham}
 	yield an isomorphism of $k[C]$-modules:
 	%
 	\begin{equation} \label{eqn:TiM=T1M_chi}